JR Hakata City
This largest train building / retail complex in Kyushu is directly connected to JR Hakata Station. It is made up of AMU Plaza Hakata, and Hakata Hankyu at the Hakata Entrance on the west side and HAKATA DEITOS and AMU EST at the Chikushi Entrance on the east side.
If you're staying near the station or coming back at night, this place is perfect—you can shop, grab a bite, and pick up souvenirs all in one go.